We study the algebraic dimension $a(X)$ of a compact hyperkähler manifold of dimension $2n$.
We show that $a(X)$ is at most
$n$ unless $X$ is projective. If a compact Kähler manifold with algebraic
dimension 0 and Kodaira dimension 0 has a minimal model,
then only the values 0, $n$ and $2n$ are possible. In case of middle
dimension, the algebraic reduction is holomorphic Lagrangian. If
$n = 2$, then - without any assumptions - the algebraic dimension
only takes the values 0, 2 and 4. The paper also gives structure results
for ”generalised hyperkähler” manifolds and studies nef lines
bundles.